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We introduce a four-factor arithmetic model for electricity baseload spot prices in Germany and Austria. The model consists of a deterministic seasonality and trend function, both short- and long-term stochastic components, and exogenous factors such as the daily wind energy production forecasts, the residual demand and the wind penetration index. We describe the short-term stochastic factor by a Levy semi-stationary (LSS) process, and the long-term component is modelled as a Levy process with increments belonging to the class of generalised hyperbolic distributions. We derive the corresponding futures prices and develop an inference methodology for our multi-factor model. The methodology allows to infer the various factors in a step-wise procedure taking empirical spot prices, futures prices and wind energy production and total load data into account. Our empirical work shows that taking into account the impact of the wind energy generation on the prices improves the goodness of fit. Moreover, we demonstrate that the class of LSS processes can be used for modelling the exogenous variables including wind energy production, residual demand and the wind penetration index.

This chapter concerns with goodness of fit tests that can be used to test whether a proposed model is consistent with data. In it the classical chi–square goodness of fit test and apply it to test for independence in contingency tables is presented. The chapter also introduces the Kolmogorov–Smirnov procedure for testing whether data comes from a specified continuous probability distribution. The classical approach to obtaining a goodness of fit test of a null hypothesis that a sample has a specified probability distribution is to partition the possible values of the random variables into a finite number of regions. The situation where each member of a population is classified according to two distinct characteristics is also considered and it is shown how to use analysis to test the hypothesis that the characteristics of a randomly chosen member of the population are independent. The chapter also provides a practice exercise and solved examples.

In this paper, we develop a function of inflation, unemployment, liquidity and real effective exchange rate by applying Autoregressive Distributed Lag (ARDL) and Artificial Neural Networks (ANN). We employ the aforementioned methods to derive the so-called Phillips curve. For the empirical objective, our primary purpose is explicitly to compare two types of the Phillips curve models obtained by ANN and the econometric methods, ARDL. Then we can check the behavior of the Phillips curve in Iran. We demonstrate that the Phillips curve for the empirical data in Iran differs slightly across ANN than econometric methods. In other words, according to the structure of Iran's economy, the ANN technique outshines the other one in terms of goodness of fit and prognosis capability. Finally, under two scenarios inflation would be forecasted in Iran up to 2025. Our findings point out that the trend of price changes in Iran would have an increasing trend in the considered period.

We consider a population in which each member has any one of k possible values. We show how to test the hypothesis that a specified set of probabilities represents the proportions of the members of the population that have each of the different possible values. We consider populations in which each member is classified as having two values, and we show how to test the hypothesis that the two values of a randomly selected member of the population are independent.

This chapter discusses about the chi-square distribution. Basically chi-square distribution is the measure of which enables us to find out the degree of degree of discrepancy between the observed and expected frequencies. In this chapter we discuss the uses of chi-square test as a test for goodness of fit, as a test for independence of attributes and homogeneity chi-square.

This paper presents an investigation about cooperation in a Public Good Game using an Agent Based Model calibrated on experimental data. Starting from the experiment proposed in Colasante and Russo (2016), we analyze the dynamic of cooperation in a Public Good Game where agents receive an heterogeneous income and choose both the level of contribution and the distribution rule. The starting point is the calibration and the output validation of the model using the experimental results. Once tested the goodness of fit of the Agent Based Model, we run some policy experiment in order to verify how each distribution rule, i.e. equidistribution, proportional to contribution and progressive, affects the level of contribution in the simulated model. We find out that the share of cooperators decreases over time if we exogenously set the equidistribution rule. On the contrary, the share of cooperators converges to 100% if we impose the progressive rule. Finally, the most interesting result refers to the effect of the progressive rule. We observe that, in the case of high inequality, this rule is not able to reduce the heterogeneity of income. This research has received funding from the European Union, Seventh Framework Programme under grant agreements n. SYMPHONY—ICT-2013-611875.

These authors congratulate Albert Maydeu-Olivares on his lucid and timely overview of goodness-of-fit assessment in IRT models, a field to which he himself has contributed considerably in the form of limited information statistics. In this commentary, Oberski and Vermunt focus on two aspects of model fit: (1) what causes there may be of misfit; and (2) what consequences misfit may have. They present their view on these topics in an integrated framework that is slightly different from that presented by Professor Maydeu-Olivares. They next provide a short illustration using an IRT analysis and, finally, draw some conclusions. This commentary expands on Maydeu-Olivares' overview of goodness-of-fit testing in item response theory by providing a model-based framework for GOF assessment. Oberski and Vermunt contend that many unresolved issues remain with the use of these measures, such as the problem of equivalent models, multiple testing, dealing with dependencies between parameters, and a shift of responsibility for deciding what may cause misfit and what is "of interest" toward the practitioner. These issues are, however, not absent in other goodness-of-fit assessment procedures, but merely less explicit. Therefore the model-based perspective appears to be a fruitful area of future research.

The distribution of microorganisms in a dilute suspension generally, though not always, accords well with a random (Poisson) distribution. However, the true distribution of organisms within foods is rarely seen because the procedures used to assess their numbers disrupt microcolonies and clumps that have resulted from cell growth or death. Yet the results of analytical tests often suggest that the distribution of organisms is over-dispersed. This chapter considers criteria and tests to assess whether the distribution of organisms conforms to a random, a regular or a contagious (over-dispersed) distribution. Such tests include the Fisher index of dispersion, the likelihood ratio index ( G ) and the goodness of fit to a particular distribution model. Over-dispersion is now recognised as being of special importance in relation to the occurrence of clumps of high numbers of food-borne pathogens (eg, Cronobacter spp.) in otherwise non-contaminated food products, such as infant formulae. Such distributions may conform to one of several distribution models including the Poisson–gamma, Poisson–lognormal, zero-inflated Poisson, etc.

An EGARCH model in which the conditional distribution is heavy-tailed and skewed is proposed. The properties of the model, including unconditional moments, autocorrelations and the asymptotic distribution of the maximum likelihood estimator, are obtained. Evidence for skewness in conditional t-distribution is found for a range of returns series and the model is shown to give a better .t than the corresponding skewed-t GARCH model., two components.

In studying various real world phenomena, we begin with a random sample of data X 1 , …, X n that represents values of some sort of a subject of interest. These measurements could represent the amount of carbon dioxide, CO 2 , in the atmosphere on a daily basis, the sizes of cancerous breast tumors, the monthly average rainfall in the State of Florida, the average monthly unemployment rate in the United States, the hourly wind forces of a hurricane, etc. In order for us to probabilistically understand the behavior of these phenomena, we need to identify the probability distribution that follows or the given data are drawn from. For example, at a certain time point we say that these data follow or come from the normal or exponential probability distribution. One of the important questions then is whether observed data are representative or follow a particular probability distribution. In fact, there is nothing we can do parametrically or statistically unless through goodness-of-fit testing identify the probability density functions, pdf , that probabilistically characterizes the behavior of the given data, the phenomenon of interest.

This article is a revision of the previous edition article by I.J. Myung, volume 4, pp. 2453–2457, © 2001, Elsevier Ltd.

This Chapter is about the techniques, formal and informal, that are commonly used to give quantitative answers in the field of distributional analysis - covering subjects including inequality, poverty and the modelling of income distributions. It deals with parametric and non-parametric approaches and the way in which imperfections in data may be handled in practice.

Spare parts are known to be associated with intermittent demand patterns and such patterns cause considerable problems with regards to forecasting and stock control due to their compound nature that renders the normality assumption invalid. Compound distributions have been used to model intermittent demand patterns; there is however a lack of theoretical analysis and little relevant empirical evidence in support of these distributions. In this paper, we conduct a detailed empirical investigation on the goodness of fit of various compound Poisson distributions and we develop a distribution-based demand classification scheme the validity of which is also assessed in empirical terms. Our empirical investigation provides evidence in support of certain demand distributions and the work described in this paper should facilitate the task of selecting such distributions in a real world spare parts inventory context. An extensive discussion on parameter estimation related difficulties in this area is also provided.

Health related Quality of Life (HrQoL) is one of the most important outcome measure in clinical trials over the past 20 years. More recently, it became also more important in epidemiological surveys, where unlike clinical trials, the number of end points involved is generally larger. In both setting, Epidemiology, or clinical trials, its measurement and statistical analysis remain an issue. The validation of a Health related Quality of Life (HrQoL) measurement is generally mainly done by internal consistency methods as external standards or experts are generally not available. These methods are mainly based on the statistical validation, using goodness of fit tests, of measurement models. We will show in this chapter, how such validation, can be done using the empirical Backward Reliability Curve (the α -curve). Finally, we present some new simulation and real data results.

In this contribution we aim at anchoring Agent-Based Modeling (ABM) simulations in actual models of human psychology. More specifically, we apply unidirectional ABM to social psychological models using low level agents (i.e., intra-individual) to examine whether they generate better predictions, in comparison to standard statistical approaches, concerning the intentions of performing a behavior and the behavior. Moreover, this contribution tests to what extent the predictive validity of models of attitude such as the Theory of Planned Behavior (TPB) or Model of Goal-directed Behavior (MGB) depends on the assumption that peoples' decisions and actions are purely rational. Simulations were therefore run by considering different deviations from rationality of the agents with a trembling hand method. Two data sets concerning respectively the consumption of soft drinks and physical activity were used. Three key findings emerged from the simulations. First, compared to standard statistical approach the agent-based simulation generally improves the prediction of behavior from intention. Second, the improvement in prediction is inversely proportional to the complexity of the underlying theoretical model. Finally, the introduction of varying degrees of deviation from rationality in agents' behavior can lead to an improvement in the goodness of fit of the simulations. By demonstrating the potential of ABM as a complementary perspective to evaluating social psychological models, this contribution underlines the necessity of better defining agents in terms of psychological processes before examining higher levels such as the interactions between individuals. © 2009 Elsevier B.V.

The Goodness of Fit (GOF) of a statistical model describes how well it fits into a set of observations. GOF indices summarize the discrepancy between the observed values and the values expected under a statistical model. GOF statistics are GOF indices used in statistical hypothesis testing. We discuss GOF statistics and indices in the context of multivariate discrete and well as multivariate continuous data.

The BHEP (Baringhaus-Henze-Epps-Pulley) test for assessing univariate and multivariate normality has shown itself to be a relevant test procedure, recommended in some recent comparative studies. It is well known that the finite sample behaviour of the BHEP goodness-of-fit test strongly depends on the choice of a smoothing parameter h. A theoretical and finite sample based description of the role played by the smoothing parameter in the detection of departures from the null hypothesis of normality is given. Additionally, the results of a Monte Carlo study are reported in order to propose an easy-to-use rule for choosing h. In the important multivariate case, and contrary to the usual choice of h, the BHEP test with the proposed smoothing parameter presents a comparatively good performance against a wide range of alternative distributions. In practice, if no relevant information about the tail of the alternatives is available, the use of this new bandwidth is strongly recommended. Otherwise, new choices of h which are suitable for short tailed and long tailed alternative distributions are also proposed.

This chapter continues with an explanation of the concept of goodness of fit for comparing two columns of data, X and Y, using linear regression. The topic described includes linear regression and the concept of computing a confidence limit for correlation.

Goodness-of-fit measures for latent variable models for binary responses are discussed. Overall goodness-of-fit statistics such as the Pearson chi-squared test and the likelihood ratio test can only be used when the observed and expected frequencies under the model are large enough. When sparseness is present, limited information statistics that use information from the lower-order margins have been proposed in the literature. Those statistics and a new one that is based on the odds ratio are presented and compared in terms of Type I error and their statistical power. Simulated results and a real example are used for exploring the performance of the overall and limited information goodness-of-fit tests under different number of items, sample size and degree of sparseness. Standardized and adjusted residuals are also studied.

Publisher Summary This chapter discusses Karl Pearon's paper on the chi square goodness of fit test. This paper presents the first mathematical account for a goodness of fit test that could be used for any shape curve. Together with other papers of the time by Pearson and colleagues, it raised the practice of mathematical statistics. The statistical framework of Pearson's chi-square is a tripartite system, for it not only incorporates a probability distribution and a goodness of fit test, but it also includes a statistical technique that Pearson introduced in when he extended his goodness of fit test to manifold classifications for the analysis of contingency tables. This technique is “chi-square contingency coefficient” to test differences between observed cell frequencies and theoretically expected cell frequencies. The development of Pearson's work on curve-fitting and finding a goodness of fit test for asymmetrical distributions may be seen as his reaction to the status accorded to the normal distribution at the end of the 19th century. Throughout the 20th century, Pearsonian statistics exerted a major influence on the development of various disciplines and continues to play a pivotal role in industry, education and in the biological, behavioral, medical, social, and human sciences.

Publisher Summary This chapter deals with the analysis of data on multiple events when the available data are limited to counts of the number of events that occur between clinic visits. The focus is on model appraisal. The choice of appropriate explanatory variables in a regression model is a major component of model appraisal. However, methods for this are widely known. Therefore, the analyses reported in the chapter incorporate the risk factors found by Gladman et al. without further investigation of this aspect of the models. A Markov model based on a categorization of the count data is examined. Perez-Ocon et al. used a Markov process to model the influence of treatments in relapse and survival times to breast cancer. Keiding et al. used such models to analyze the course of events after bone marrow transplantation. Most goodness of fit tests for Markov models assume that, for each subject, a response variable is recorded at a fixed number of times and that no explanatory variables are measured. The inter-visit counts are modeled directly with Poisson and negative binomial distributions and the need for special treatment of zero counts in such models is examined.

The assumption of univariate and multivariate normality is implicit in most of the statistical procedures routinely used in the analysis of univariate and multivariate data. Now it is well recognized that, in general, the assumption of normality is at best suspect; e.g., see Geary (1947) , Pearson (1929) , Jeffreys (1961) , Mudholkar and Srivastava (2000a) and references therein. Furthermore, it is also well established that when the assumption of normality is violated most of the normal theory procedures lose validity, i.e., Type I error control, or become highly inefficient in terms of power. Numerous goodness-of-fit methods to test the assumption of univariate normality exist in the literature but no single test uniformly dominates all others. However, several theoretical and simulation justifications published in the literature indicate that the Shapiro-Wilk test is reasonable and appropriate in most situations of practical importance. The assumption of multivariate normality is harder to expect and justify since it implies joint normality, in addition to the marginal normality, of the components. This structural complexity may be a reason for a time lag in the development of goodness-of-fit tests for multivariate normality. However, the last two decades have seen advances leading to several competing tests of multivariate normality. In addition, it is seen that, as compared with the univariate methods, the multivariate data analysis methods are more prone to becoming invalid in terms of Type I error control and inefficient in terms of power when the normality assumption is violated. The purpose of this article is to present an overview of the methods for testing univariate and multivariate normality and to indicate their relative strengths and weaknesses.

Publisher Summary This chapter discusses goodness-of-fit tests for the Cox proportional hazards model that are based on ideas similar to the Hosmer and Lemeshow goodness-of-fit test for logistic regression. All of these tests can be derived by adding group-indicator variables to the model and testing the hypothesis that the coefficients of the group indicator variables are zero via the score test. The tests that can be derived in this way are called the “added variable tests.” Care needs to be taken when implementing these tests because some of them require the use of time-dependent group indicator variables. Information regarding the time-dependent nature of the tests is also provided along with examples.

Goodness of fit pertains to statistical model assessment: it asks the question of how well a probability model, such as the normal distribution, agrees with observed data. The first objective assessment of this question goes back to Karl Pearson, writing at the end of the nineteenth century. His famous criterion involves summing over all categories the quantity observed count minus the expected count all squared and divided by the expected count. Modern approaches to this question focus on statistics using the empirical distribution function, and on the smooth tests.

Assessing the goodness of fit of a statistical model involves comparisons of actual data with those expected under the model. This process of such assessment often involves formal comparisons between models as well, and thus shares a number of features with methods associated with the problem of model selection from a class of parametric models. This entry describes the general philosophy associated with these dual problems of goodness of fit and model choice, and discusses some of the basic approaches to them from both a frequentist and a Bayesian perspective. Trying to find the best model may not prove possible in all circumstances, but the array of methods currently available often allow statistical analysts to identify parsimonious models whose fit to the observed data is sufficient for the purpose at hand.

Publisher Summary The comparison of different hypotheses, i.e. of competing models, is the basis of model specification. It may be performed along two main lines. The first one consists in associating with each model a loss function and in retaining the specification implying the smallest (estimated) loss. In practice, the loss function is defined either by updating some a priori knowledge on the models given the available observations (the Bayesian point of view), or by introducing some criterion taking into account the trade-off between the goodness of fit and the complexity of the model. The second approach is hypothesis testing theory. However, the determination of the decision rule is not done on the same basis as model choice. The basis of hypothesis testing theory is to introduce the probability of errors. This chapter focuses on the case where none of the hypotheses is a particular case of another one.

Publisher Summary When considering residuals, a qualitative approach is often the most revealing and informative. Though, quantitative methods to test more rigorously particular properties of residuals must be considered sometimes to quantitate the statistical significance of conclusions drawn as a result of data analysis. This chapter discusses some of the available methods for considering residuals with the aid of illustrative examples. The definition of an appropriate mathematical model involves considerations of the way to transform the experimentally observed system behavior into a mathematical description that permits a physical interpretation of the model parameter values. The significance of subtle behavior in residuals may suggest the presence of a significant system property that is overlooked by the current mathematical model. After verification that the residuals resulting from a model parameter estimation process to a set of data are indeed free of any systematic trends relative to any variables of significance (that is, dependent or independent variables), a quantitative estimate of the adequacy of the particular model in describing the data is possible. The concepts and approaches outlined in the chapter provide a survey of methods available for qualitatively and quantitatively considering residuals generated from data analysis procedures. In those cases where very precise interpretation of experimental observations is required, a thorough quantitative consideration of residuals may be necessary to address the statistical validity of particularly detailed mathematical models designed to account for the biochemical or biophysical properties of any experimental system of interest.

This chapter presents the analysis of variance. A method of testing for significance differences between means and proportions where more than two samples are involved is given by the analysis of variance. It permits the testing of the statistical significance between the means and proportions derived from several samples. The analysis of variance proceeds by separating or partitioning the total variance in the data into two parts: (1) the variance between the samples and (2) the variance within the individual samples. Two estimates of the population variance are then calculated: (1) one based on the variance between one sample and another, and (2) the other based on the variation of the values between each sample. When the analysis of variation is completed, the two estimates are compared with each other, and the ratio between them is evaluated for statistical significance.

Publisher Summary This chapter presents two studies to discuss the robustness of item response models and effects of test length and sample size on the precision of ability estimates. The purpose of Study 1 was to study systematically the goodness of fit of the one-, two-, and three-parameter logistic models. Using computer-simulated test data, the effects of four variables were studied: (1) the variation in item discrimination parameters; (2) the average value of the pseudochance-level parameters; (3) the test length; and (4) the shape of the ability distribution. Artificial or simulated data representing the departures of varying degrees from the assumptions of the three-parameter logistic test model were generated and the goodness of fit of the three test models to the data was studied. The goodness-of-fit measures used were chosen for their practical significance. The Study 2 was designed to investigate two practical questions that are important to test developers: (1) the effects of examinee sample size and test length on the standard errors of ability estimation (SEE) curves and (2) the effects that the statistical characteristics of an item pool have on the precision of standard errors of ability estimation curves. A study of the use of SEE curves and factors that affect their stability was motivated by item response model test development procedures. When item statistics are available they are commonly used by test developers for selecting test items from a pool of test items to produce a test with a desired SEE curve.

Publisher Summary A study purporting to show the growth of mental ability, as measured by the Binet test, as a function of chronological age was published in 1929 by Thurstone and Ackerson. The curve was based on a rescaling of Binet mental ages (MA) in a cross-sectional sample of 4208 children age 3 through 17 seen at the Institute for Juvenile Research in Chicago. Any discussion of the shape of such curves requires that the unit of scale be equal at all points throughout the range of measurement. This chapter discusses a scaling procedure fully independent of chronological age units. With the aid of modern computers, however, the Binet test, or similar tests to chronological age or to another external criterion, can be scaled on the single assumption that the underlying distributions for item attainments conditional on age have a known common functional form indexed by a threshold and a dispersion parameter. If this assumption is satisfied, scale values may be assigned to the chronological age groups so that with respect to the growth continuum, all the item response curves simultaneously fit the observed percentage-correct data for each item in each age group. On the further assumption that item responses within the age groups are locally independent, the goodness of fit of the solution can be tested by a large-sample statistical test.